+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1 +o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
3. Let n be an integer. Prove that 2 (n4 – 3) if and only if 4| (n2+3).
Let 1 ≤ p ≤ q and suppose that for 1 ≤ j ≤ n we have xj ≥ 0. Prove that ( n ∑ j=1 (xj^q) )^ (1/q) ≤ ( n ∑ j=1 (xj^p) )^ (1/p) ≤ n^ (1/p - 1/q) ( n ∑ j=1 (xj^q) )^ (1/q)
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...
real analysis proof Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1 Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1
(4) Let(an}n=o be a sequence in C. Define R-i-lim suplanlì/n. Recall that R e [0,x] o0 is the radius of convergence of the power series Σ a (z 20)" Assume that R > 0 (a) Prove that if 0 < ρ < R, then the power series converges uniformly on the closed (b) Prove that the power series converges uniformly on any compact subset of the disk Ix - xo< R (4) Let(an}n=o be a sequence in C. Define R-i-lim...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R. Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
Use induction to prove that 0–0 4j3 = n4 + 2n3 + n2 where n > 0.
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo (12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young's inequality with E. Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young's inequality with E.
(n) 6. Let X ~ Binomial (n,p). Prove that a. Ex=0 (6)p*(1 – p)n-* = ... = 1 b. E[X] = 21-0 x()p*(1 - 2)^-^ = = mp c. Var[X] = x=0x2 (1)p*(1 – p)n-x – (np)2 = ... = np(1 – p) d. My(t) = ... = (pet + 1 - p)n